Tilting modules under special base changes

Given a non-unit, non-zero-divisor, central element x of a ring Λ, it is well known that many properties or invariants of Λ determine, and are determined by, those of Λ/xΛ and Λx. In the present paper, we investigate how the property of “being tilting” behaves in this situation. It turns out that any tilting module over Λ gives rise to tilting modules over Λx and Λ/xΛ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over Λ is tilting if its corresponding localization and quotient are tilting over Λx and Λ/xΛ respectively.